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Long Division

RemainderTheorem

Factor Theorem

SyntheticDivision

Rational Zeros

Theorem

Upper &Lower

Bounds

Division involves a dividend divided by a divisor to obtain a

quotient and a _________.

remainder

What is the answer of 2x4-x3-2 divided by 2x2+x+1? Write in

polynomial form.

2x4-x3-x=(2x2+x+1)(x2-2)+(x-2)

Divide f(x)=x2-2x+3 b x-1. Write in polynomial form.

f(x)=(x=1)2+2

f(x)=(x2+x+4)(x+3)-21

Divide f(x)=x3+4x2+7x-9 by x+3 and write a summary statement

in polynomial form.

Divide f(x)=x4-2x3+3x2-4x+6 by x2+2x-, write a summary

statement in fraction form.

f(x)=(x2-4x+12)(x2+2x-1)-32x+18

If a polynomial is divided by ___, then the remainder is

r=f(k).

x-k

Using the remainder theorem, find the remainder when

f(x)=3x2+7x-20 is divided by x+1.

r=f(-1)=3(-1)2+7(-1)-20=3-7-20=-24

What is the remainder when f(x)=2x2-3x+1 is divided by x-2?

3

Use the remainder theorem to find the remainder when you divide f(x)=x3-x2+2x-1 by x+3

-43

5

Use the remainder theorem to find the remainder when f(x)=2x3-3x2+4x-7 by x-2

0

A polynomial function f(x) has a factor x-k if and only if f(k)=_.

(3x-5)

Using the factor theorem, factor f(x)=3x2+7x-20 by dividing it by

the known factor x+4.

Yes

Is x-1 a factor of x3-x2+x-1?

No

Use the factor theorem to determine whether x-2 is a

factor of x3+3x-4

Yes

Use the factor theorem to determine where x+2 is a factor

of 4x3+9x2-3x-10

linear divisor

This shortcut method for the division of a polynomial by a

______ _______ x-k is synthetic division.

2x3-3x2-5x-12 = 2x2+3x+4

x-3

Divide 2x3-3x2-5x-12 by x-3 using synthetic division and

write a summary statement in fraction form.

X2-6x+9+ -11

x+1

Divide using synthetic division and write a summary statement

in fraction form: x3-5x2+3x-2

x+1

9x2+97x+967+ 9670

x-10

Divide using synthetic division and write a summary statement

in fraction form: 9x3+7x2-3x

x-10

-5x3-20x2-80x-317+ -1269

4-x

Divide using synthetic division and write a summary statement

in fraction form: 5x4-3x+1

4-x

rational, irrational

Real zeros of a polynomial function are either ________

zeros – zeroes that are rational numbers – or __________

zeros – zeros that are irrational numbers.

f(1)=(1)3-3(1)2+1=-1

f(-1)=(-1)3-3(-1)2+1=-3

So f has no rational zeros.

Find the rational zeros of f(x)=x3-3x2+1

1

1, 2, 3, 6

(all positive & negative)

Use the rational zeroes theorem to write a list of all possible

rational zeroes of f(x)=6x3-5x-1

1, 3, 9

1, 2

(all positive & negative)

Use the rational zeroes theorem to write a list of possible rational

zeroes:f(x)=2x3-x2-9x+9

-1/2 and 4; rational

No irrational

Find all the zeroes of the function and identify each zero

as rational or irrational:f(x)=2x4-7x3-2x2-7x-4

bounds

We narrow our search for real zeros by using a test that identifies upper and lower

______ for real zeros.

-2

Find a lower bound of f(x)=2x4-7x3-8x2+14x+8

Yes

Use synthetic division to prove that to number k is an upper

bound for the real zeroes of the function:

k=3; f(x)=2x3-4x2+x+3

Yes

Use synthetic division to prove that the number k is an upper

bound for the real zeroes of the function f:

k=2; f(x)=x4-x3+x2+x-12

Yes

Use synthetic division to prove that the number k is a lower

bound for the real zeroes of the function f:

k=-1; f(x)=3x2-4x2+x-2